Numerical method for unsteady viscous hydrodynamical problem with free boundaries

A finite element method for the transient incompressible Navier–Stokes equations with the ability to handle multiple free boundaries is presented. Problems of liquid–liquid type are treated by solving two coupled Navier–Stokes problems for two separate phases. The possibility to solve problems of liquid–gas, liquid–liquid–gas or liquid–liquid–liquid type is demonstrated too. Surface tension effects are included at deformable interfaces. The method is of Lagrangian type with mesh redefinition. A predictor‐corrector scheme is used to compute the position of the deformable interface with automatic control of its accuracy and smoothness. The method is provided with an automatic choice of the time integration step and an optional spline filtration of the truncation error at the free surface. In order to show the accuracy of the method, tests and comparisons are presented. Numerical examples include motion of bubbles and multiple drops.